The approximation property and exactness of locally compact groups. (English) Zbl 1465.22003
As mentioned in the abstract of the paper by Suzuki, the theorem of Haagerup and Kraus is extended to the \(C^*\)-algebra context. Namely, the reduced crossed product of a \(C^*\)-algebra with the (strongly) stable approximation property by an action of a locally compact group with the approximation property has the same property preserved. Solved by its method is an open unapprochable implication that the approximation property of a general locally compact group implies the group to become exact.
We may recall the following for convenience if to be started or not.
A locally compact group is said to have the approximation property if there is a net of the Fourier group algebra converging to the constant one function on the group in the certain \(\sigma\)-topology.
A \(C^*\)-algebra is said to have the (strongly) stable approximation property if the identity map of the \(C^*\)-algebra is in the closure of finite rank operators as in completely bounded linear maps of the \(C^*\)-algebra in the (strongly) stable point-wise norm topology, tensored with the identity map on the \(C^*\)-algebra of all compact operators (or of all bounded operators) on an infinite-dimensional Hilbert space.
As a discovery point of view, it seems to be from discrete to general, as a converse of some recent current from continuous to discrete.
We may recall the following for convenience if to be started or not.
A locally compact group is said to have the approximation property if there is a net of the Fourier group algebra converging to the constant one function on the group in the certain \(\sigma\)-topology.
A \(C^*\)-algebra is said to have the (strongly) stable approximation property if the identity map of the \(C^*\)-algebra is in the closure of finite rank operators as in completely bounded linear maps of the \(C^*\)-algebra in the (strongly) stable point-wise norm topology, tensored with the identity map on the \(C^*\)-algebra of all compact operators (or of all bounded operators) on an infinite-dimensional Hilbert space.
As a discovery point of view, it seems to be from discrete to general, as a converse of some recent current from continuous to discrete.
Reviewer: Takahiro Sudo (Nishihara)
MSC:
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 46L05 | General theory of \(C^*\)-algebras |
| 46L55 | Noncommutative dynamical systems |
Keywords:
approximation property; exactness; locally compact group; crossed product; group C*-algebra; tensor productReferences:
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